Condensed matter is the largest, most diverse field of physics, and perhaps the one with the most immediate impact on our everyday lives. Electronic structure theory is playing an ever increasing role in this field, as both our computers and our abilities to create complex materials are constantly improving.

For materials, there are two fairly distinct approaches. The more traditional is to write some model Hamiltonian that describes the most important physics, fit its parameters to experimental information, and find accurate (if not exact) solutions to this Hamiltonian. This remains the dominant paradigm in strongly correlated systems, where even a qualitative understanding of some exotic phenomena are missing such as for the cuprate and pnictide superconductors.

The other approach is to take the full electronic Hamiltonian, and attempt to accurately solve the quantum mechanics without empirical input. This is called first-principles, and often the primary aim is to find the ground-state energy extremely accurately, as very tiny energy differences determine, for example, where molecules will adsorb on a surface or how fast a chemical reaction will occur. This pursuit is shared by quantum chemists for molecules and computational scientists working on materials, and even some in geoscience.

The starting point of most such investigations is Kohn-Sham density functional theory (DFT), and Richard Martin's book Electronic Structure: Basic Theory and Practical Methods (2004) is an indispensable guide to this subject. Density functional theory is a tremendous physics success story, but relatively underappreciated in its home community. For example, density functional theory calculations were behind the prediction1 and subsequent finding2 of the world's highest temperature (203 K) superconductor, hydrogen sulfide under pressure. These calculations are now so widespread that John Perdew, the condensed matter theorist responsible for much of modern density functional theory development, is substantially more cited than Ed Witten, Phil Anderson, or Walter Kohn. (In fact, Perdew appears to be the most cited physicist ever.)

But most physicists, while they need to know where the atoms are and what reactions will occur, care more about the response of a material to external stimuli (photons, electrons, etc.) than to, e.g., its cohesive energy. Basic density functional theory is designed to yield only ground-state properties. Moreover, the popular approximations in density functional theory tend to fail for strongly correlated systems, which include many of the oxide materials vital to energy technologies. There is an extreme need to both extract response properties and to develop more accurate, reliable alternatives to vanilla density functional theory. Moreover, density functional theory takes such a radically different approach from the outset that it is almost impossible to connect it with standard approaches to solving the Schrödinger equation. (The simple two-site Hubbard model can be a good starting point.3)

This is the great value of Interacting Electrons: Theory and Computational Approaches. It provides an overview of the interacting electron problem, but also lays a fairly complete introduction to density functional theory and Green's functions. The main purpose of this book is to go well beyond density functional methods and in doing so it provides the foundations of the GW approximation, the Bethe-Salpeter equation, Dynamical Mean Field Theory, quantum Monte Carlo, and much more. There is significant breadth in these topics, but Interacting Electrons does a noteworthy job of introducing them in a sensible order and paying homage to the influences and differing approaches from which they arose. This may be the only volume in which all these topics are simultaneously treated with sufficient depth and in a shared framework—laying the conceptual basis, showing the actual methods (often with handy flowcharts), and demonstrating various applications, including descriptions of areas within condensed matter that have benefited from these methods. The appendices are also of very high quality and are an indispensable tool for bringing readers up to speed.

We recommend this book strongly for at least three disparate audiences. Because of the excellent treatment of fundamentals, this should be required reading for a second or third year graduate student who runs materials calculations but also wants a deeper understanding of the underlying theory. It is also a must-read for anyone working in strongly-correlated systems who needs to understand the performance and limitations of these many approaches to electronic structure. And while this is certainly not a book about molecules, with only passing references at best, it does form an excellent self-contained starting point for quantum chemists who are working on materials problems and wish to learn the strengths and weaknesses of materials methods and codes.

Finally, this book provides a great reference for all those folks who are not running such calculations, but need to read and understand the papers of those who are.

1.
D.
Duan
,
Y.
Liu
,
F.
Tian
,
D.
Li
,
X.
Huang
,
Z.
Zhao
,
H.
Yu
,
B.
Liu
,
W.
Tian
, and
T.
Cui
, “
Pressure-induced metallization of dense (H2S)2H2 with high-Tc superconductivity
,”
Sci. Rep.
4
,
6968
(
2014
).
2.
A. P.
Drozdov
,
M. I.
Eremets
,
I. A.
Troyan
,
V.
Ksenofontov
, and
S. I.
Shylin
, “
Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system
,”
Nature
525
,
73
76
(
2015
).
3.
D. J.
Carrascal
,
J.
Ferrer
,
J. C.
Smith
, and
K.
Burke
, “
The Hubbard dimer: A density functional case study of a many-body problem
,”
J. Phys.: Condens. Matter
27
,
393001
(
2015
).

Justin C. Smith is an NSF Graduate Research Fellow in the Department of Physics at University of California, Irvine. He does research on thermal density functional theory and its applications to warm dense matter.

Kieron Burke is a Chancellor's Professor of Chemistry and of Physics at University of California, Irvine, and has worked on electronic structure theory for about 25 years.